In welfare economics, the fundamental property of concave welfare functions is that they are monotonic with respect to the dominance of generalized Lorenz curves [1]}, [2]}, [3]}, because this garantees that maximizing a welfare function performs an optimal redistribution from the better-off to the worse-off at every level of average utility. In the context of two-sided fairness in rankings, [4]} formalize their fairness criterion by stating that a ranking policy is fair as long as the generalized Lorenz curves of users and items are not jointly dominated. In this section, we show that the family of GGFs \(F_{\lambda , {w^1}, {w^2}}(P)\)  (REF ) allows to generate every ranking policy that are fair under this definition, and only those. The result follows from standard results of convex multi-objective optimization [5]}, [6]}. We give here the formal statements for exhaustivity.
